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## 生物代考|生物统计学代考BIOSTATISTICS代考|BIOL220 The Sample Size for Simple and Systematic Random Samples

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## 生物代考|生物统计学代考BIOSTATISTICS代考|The Sample Size for Simple and Systematic Random Samples

In a simple random sample or a systematic random sample, the sample size required to produce a prespecified bound on the error of estimation for estimating the mean is based on the number of units in the population $(N)$, and the approximate variance of the population $\sigma^{2}$. Moreover, given the values of $N$ and $\sigma^{2}$, the sample size required for estimating a mean $\mu$ with bound on the error of estimation $B$ with a simple or systematic random sample is
$$n=\frac{N \sigma^{2}}{(N-1) D+\sigma^{2}}$$
where $D=\frac{B^{2}}{4}$. Note that this formula will not generally return a whole number for the sample size $n$; when the formula does not return a whole number for the sample size, the sample size should be taken to be the next largest whole number.
Example $3.11$
Suppose a simple random sample is going to be taken from a population of $N=5000$ units with a variance of $\sigma^{2}=50$. If the bound on the error of estimation of the mean is supposed to be $B=1.5$, then the sample size required for the simple random sample selected from this population is
$$n=\frac{5000(50)}{4999\left(\frac{1.52}{4}\right)+50}=87.35$$
Since $87.35$ units cannot be sampled, the sample size that should be used is $n=88$. Also, $n=$ 88 would be the sample size required for a systematic random sample from this population when the desired bound on the error of estimation for estimating the mean is $B=1.5$. In this case, the systematic random sample would be a 1 in 56 systematic random sample since $\frac{5000}{88} \approx 56$.

## 生物代考|生物统计学代考BIOSTATISTICS代考|The Sample Size for a Stratified Random Sample

Recall that a stratified random sample is simply a collection of simple random samples selected from the subpopulations in the target population. In a stratified random sample, there are two sample size considerations, namely, the overall sample size $n$ and the allocation of $n$ units over the strata. When there are $k$ strata, the strata sample sizes will be denoted by $n_{1}, n_{2}, n_{3}, \ldots, n_{k}$, where the number to be sampled in strata 1 is $n_{1}$, the number to be sampled in strata 2 is $n_{2}$, and so on.

There are several different ways of determining the overall sample size and its allocation in a stratified random sample. In particular, proportional allocation and optimal allocation are two commonly used allocation plans. Throughout the discussion of these two allocation plans, it will be assumed that the target population has $k$ strata, $N$ units, and $N_{j}$ is the number of units in the $j$ th stratum.

The sample size used in a stratified random sample and the most efficient allocation of the sample will depend on several factors including the variability within each of the strata, the proportion of the target population in each of the strata, and the costs associated with sampling the units from the strata. Let $\sigma_{i}$ be the standard deviation of the $i$ th stratum, $W_{i}=N_{i} / N$ the proportion of the target population in the $i$ th stratum, $C_{0}$ the initial cost of sampling, $C_{i}$ the cost of obtaining an observation from the $i$ th stratum, and $C$ is the total cost of sampling. Then, the cost of sampling with a stratified random sample is
$$C=C_{0}+C_{1} n_{1}+C_{2} n_{2}+\cdots+C_{k} n_{k}$$
The process of determining the sample size for a stratified random sample requires that the allocation of the sample be determined first. The allocation of the sample size $n$ over the $k$ strata is based on the sampling proportions that are denoted by $w_{1}, w_{2}, \ldots w_{k}$. Once the sampling proportions and the overall sample size $n$ have been determined, the $i$ th stratum sample size is $n_{i}=n \times w_{i}$.

The simplest allocation plan for a stratified random sample is proportional allocation that takes the sampling proportions to be proportional to the strata sizes. Thus, in proportional allocation, the sampling proportion for the $i$ th stratum is equal to the proportion of the population in the $i$ th stratum. That is, the sampling proportion for the $i$ th stratum is
$$w_{i}=\frac{N_{i}}{N}$$

## 生物代考|生物统计学代考BIOSTATISTICS 代考|The Sample Size for Simple and Systematic Random Samples

$$n=\frac{N \sigma^{2}}{(N-1) D+\sigma^{2}}$$

$$n=\frac{5000(50)}{4999\left(\frac{1.52}{4}\right)+50}=87.35$$

## 生物代考|生物统计学代考BIOSTATISTICS 代考|The Sample Size for a Stratified Random Sample

$$C=C_{0}+C_{1} n_{1}+C_{2} n_{2}+\cdots+C_{k} n_{k}$$

$$w_{i}=\frac{N_{i}}{N}$$

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